mlsanityData to 14 Oct 2026

Graph Neural Preconditioners for Iterative Solutions of Sparse Linear Systems

arXiv 2406.00809 ↗First posted 2 Jun 2024math.NA cs.LG cs.NAcited by 15

Not on the map. Why?

Abstract (arXiv)

Preconditioning is at the heart of iterative solutions of large, sparse linear systems of equations in scientific disciplines. Several algebraic approaches, which access no information beyond the matrix itself, are widely studied and used, but ill-conditioned matrices remain very challenging. We take a machine learning approach and propose using graph neural networks as a general-purpose preconditioner. They show attractive performance for many problems and can be used when the mainstream preconditioners perform poorly. Empirical evaluation on over 800 matrices suggests that the construction time of these graph neural preconditioners (GNPs) is more predictable and can be much shorter than that of other widely used ones, such as ILU and AMG, while the execution time is faster than using a Krylov method as the preconditioner, such as in inner-outer GMRES. GNPs have a strong potential for solving large-scale, challenging algebraic problems arising from not only partial differential equations, but also economics, statistics, graph, and optimization, to name a few.

Key facts. Cited by 15 papers in the map (data through 14 Oct 2026). Builds on 23 of its 52 references that are in the map. First posted on arXiv 2 Jun 2024.

Builds on

23 of 52 references.

References · 23

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  2. 2017Graph Attention Networkscited by 8,923
  3. 2017Inductive Representation Learning on Large Graphscited by 6,320
  4. 2018How Powerful are Graph Neural Networks?cited by 4,393
  5. 2021On the Opportunities and Risks of Foundation Modelscited by 4,033
  6. 2019A Comprehensive Survey on Graph Neural Networkscited by 2,919
  7. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  8. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  9. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,576
  10. 2019Simplifying Graph Convolutional Networkscited by 1,477
  11. 2023A Comprehensive Survey of Continual Learning: Theory, Method and Applicationcited by 872
  12. 2020Simple and Deep Graph Convolutional Networkscited by 860
11 more
  1. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  2. 2018FastGCN: Fast Learning with Graph Convolutional Networks via Importance Samplingcited by 584
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  4. 2023Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-trainingcited by 265
  5. 2021Accelerating Training and Inference of Graph Neural Networks with Fast Sampling and Pipeliningcited by 39
  6. 2018Stochastic Gradient Descent with Biased but Consistent Gradient Estimatorscited by 18
  7. 2023Neural incomplete factorization: learning preconditioners for the conjugate gradient methodcited by 18
  8. 2023Communication-Efficient Graph Neural Networks with Probabilistic Neighborhood Expansion Analysis and Cachingcited by 16
  9. 2024Learning from Linear Algebra: A Graph Neural Network Approach to Preconditioner Design for Conjugate Gradient Solverscited by 15
  10. 2017Linear-Cost Covariance Functions for Gaussian Random Fieldscited by 4
  11. 2024Ising on the Graph: Task-specific Graph Subsampling via the Ising Modelcited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 8; 2026: 6.

15 papers.

Sort:

By area:

  1. 2024Learning incomplete factorization preconditioners for GMREScited by 9
  2. 2025PEARL: Preconditioner Enhancement through Actor-critic Reinforcement Learningcited by 3
  3. 2026torch-sla: Differentiable Sparse Linear Algebra with Adjoint Solvers and Sparse Tensor Parallelism for PyTorchcited by 3
  4. 2025Message-Passing GNNs Fail to Approximate Sparse Triangular Factorizationscited by 2
  5. 2025MD-PNOP: Equation-Recast Neural Operators for Minimal-Data Extrapolation and PDE Solver Accelerationcited by 2
  6. 2025Preconditioned Deformation Gridscited by 1
  7. 2025Learning Sparse Approximate Inverse Preconditioners for Conjugate Gradient Solvers on GPUscited by 1
  8. 2026Neu-PiG: Neural Preconditioned Grids for Fast Dynamic Surface Reconstruction on Long Sequencescited by 1
  9. 2026Equation Recast for Canonical Operator Learning Across Parametric PDEscited by 1
  10. 2025Accelerated Portfolio Optimization and Option Pricing with Reinforcement Learningcited by 0
  11. 2025Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theorycited by 0
  12. 2025Fast Linear Solvers via AI-Tuned Markov Chain Monte Carlo-based Matrix Inversioncited by 0
3 more
  1. 2026A level-wise training scheme for learning neural multigrid smoothers with application to integral equationscited by 0
  2. 2026RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Correctionscited by 0
  3. 2026Graph Neural Multilevel Preconditioners for Iterative Solverscited by 0